{"id":"openai-math/The-Euclidean-plane-is-not-five-colorable","pdf":"https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026","tags":["discrete-geometry","chromatic-number","combinatorics","plane-coloring"],"score":0,"title":"The Euclidean plane is not five-colorable","domain":"Combinatorics","family":"158","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":3,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://blog.computationalcomplexity.org/2026/10/conjecture-that-i-thought-would-not-be.html#:~:text=For%20nearly%2070%20years%2C%20all%20that%20was%20known%20was","kind":"web","site":"blog.computationalcomplexity.org","text":"For nearly 70 years, all that was known was \\( 4\\le \\chi\\le 7\\). We denote \\(\\chi(G)\\) by \\(\\chi\\). The proof that \\(4 \\le\\chi\\) is easy. The proof that \\(\\chi\\le 7\\) is easy. In 2018, Aubrey de Grey improved the lower bound to 5 . The proof that \\(5\\le \\chi\\) is hard: it involves a 1,581-vertex…","added":"2026-11-03","title":"Conjectures that I thought would not be proved in my lifetime....were","author":"Bill Gasarch"},{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=raises%20the%20lower%20bound%20to%20six%2C%20leaving%20six%20and%20seven%20as%20the%20two%20possible%20answers","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"raises the lower bound to six, leaving six and seven as the two possible answers. The Kakeya problem. My post on Hong Wang and Joshua Zahl’s breakthrough discussed the three-dimensional Kakeya set conjecture and the stronger maximal-function problem. The release announces the maximal conjecture in…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans","author":"Gil Kalai"},{"url":"https://x.com/DmitryRybin1/status/2107852898617229724","date":"2026-10-07","kind":"x","added":"2026-11-03","author":"Dmitry Rybin","handle":"DmitryRybin1"}],"formalized":true,"family_title":"The Euclidean plane cannot be colored with five colors"}